76 problems

Small-limit conjecture. The category $n\operatorname{Pr}^L$ has all small limits for all $n\geq 0$.

Let $\mathcal{V}$ be a presentable, closed monoidal $\infty$-category. Write $\operatorname{Cat}^{\mathcal{V}}$ for the $\infty$-category of $\mathcal{V}$-enriched categories and…

Let $M$ and $N$ be smooth connected manifolds. Write $\operatorname{{\sf{Ran}}}^{\sf{u}}(M)$ for the unital Ran space of $M$, and let $\operatorname{{\sf{Exit}}}(X)$ denote the exi…

Let $[T_{\rm poly}^{\bullet}]_{\rm div}$ denote the divergence-compatible polyvector fields, let $\widetilde{\mathcal C}$ be the explicitly constructed $L_{\infty}$-morphism before…

Let $T_{\rm poly}^{\bullet}$ be the polyvector-field Lie algebra with divergence differential $d_{\rm div}$, and let $[\mathcal D_{\rm poly}^{\bullet}]_{\rm cycl}$ be the cyclic Ho…

Characterization conjecture. The infinity-topoi are exactly the Segal categories of simplicial presheaves on Grothendieck sites.

Let $X$ be a homotopy semigroup in the category of based topological spaces or chain complexes, with underlying object $X(1)$. In the topological case, the associated structure is…

Homotopical homological mirror symmetry conjecture. The $A_\infty$ categories

Let $(\bDelta^n)$ denote the oriented simplex, let $\tau_{n-1}(\bDelta^n)$ denote its $(n-1)$-truncation, and let $(\Delta^n)^t$ be the stratified simplicial space obtained from th…

Let $\Catn$ denote the category of $(\infty,n)$-categories, let $\Shv(\grid)\univ$ denote the category of univalent sheaves on the grid, and let … be the cubical nerve. For $n=2$,…

Let $R\colon \Cat^L_\infty\hookrightarrow \Cat_\infty$ be the embedding of the category of presentable $\infty$-categories and colimit-preserving functors into the category of…

Cartesianity conjecture. If $C$ is an infinity-category, then

Constructibility conjecture. The category of categories should be constructible in $\STT{}$ as a certain subtype of the universe.

Polynomial comonad construction conjecture. Every polynomial comonad over $\mathcal{S}$ admits such a canonical construction, and every complete Segal space arises from this constr…

Let $\mathcal{S}$ be an abstract wrapped Floer setup, let $\mathcal{F}^{\text{pre}}_{\mathcal{S}}$ be its pre-wrapped Fukaya $A_\infty$-category, let $\widetilde{C}$ be the set of…

Let $X$ be a $\Gamma$-scheme, let $\mathbb T_X^\Gamma$ denote its $\Gamma$-tangent $\infty$-category, and let $\mathbb L_X^\Gamma$ denote its cotangent complex. Tangent–cotangent e…

Let $(S,n,{\bf a})$ be a parametrized surface and let $R(S,n,{\bf a};k)$ be the dga generated by diagrams of $k$ strands between the corresponding collections of arcs. Let $\mu_m$…

Let $M$ be an ungroup-like abelian monoid, let $X$ be a topological space, and define $\mathrm{SP}(X,M)$ to be the free strictly commutative topological monoid generated by $X$-man…

Let $\mathcal{C}$ be an $(\infty,n)$-category, and let $\mathrm{O}_{n-1}$ and $\Theta_n$ be the indexing shapes used to define the iterated constructions…

Let $\mathcal{C}$ be a symmetric monoidal $(\infty,n)$-category, let $S$ be a finite set, and let $\underline{X}$ and $\partial\underline{X}$ be the spaces used in the construction…

Let $\mathbf{CompCat}_{\Sigma, \Pi_\text{ext}, \text{Id}}$ be the category of models of intensional Martin–Löf type theory with dependent sums, extensional dependent products, and…

Let $\mathscr{C}$ be an $(\infty,2)$-category, and let $\mathscr{C}_v$ and $\mathscr{C}_h$ denote its vertical and horizontal inclusions as double $\infty$-categories. There is a s…

Let $\mathfrak{K}_{n}$ and $\mathfrak{M}_{n}$ be the model categories of marked Kan complexes and marked complicial sets, respectively, and let $(i_{\Delta},-_{\Delta})$ be the adj…

Let $\mathcal{C}$ be a saturated $A_{\infty}$-category and let $F$ be an endofunctor of $\mathcal{C}$. The categorical entropy $h_t(F)$ is evaluated at $t=0$ to give $h_0(F)$. Alge…

Let $n\text{-}\operatorname{Cat}$ denote the category of $n$-categories, let $\operatorname{Surj}_{\omega}$ denote the class of $\omega$-surjective morphisms, and let…